decision theory under uncertainty. uncertainty consumer and firms are usually uncertain about the...

Decision theory under uncertainty

Uncertainty

Consumer and firms are usually uncertain about the payoffs from their choices. Some examples…

Example 1: A farmer chooses to cultivate either apples or pears When he takes the decision, he is uncertain

about the profits that he will obtain. He does not know which is the best choice

This will depend on rain conditions, plagues, world prices…

Uncertainty

Example 2: playing with a fair die We will win £2 if 1, 2, or 3, We neither win nor lose if 4, or 5 We will lose £6 if 6 Let’s write it on the blackboard

Example 3: John’s monthly consumption: £3000 if he does not get ill £500 if he gets ill (so he cannot work)

Our objectives in this part

Study how economists make predictions about individual’s or firm’s choices under uncertainty

Study the standard assumptions about attitudes towards risk

Economist’s jargon

Economists call a lottery a situation which involves uncertain payoffs: Cultivating apples is a lottery Cultivating pears is another lottery Playing with a fair die is another one Monthly consumption

Each lottery will result in a prize

Probability

The probability of a repetitive event happening is the relative frequency with which it will occur probability of obtaining a head on the fair-

flip of a coin is 0.5 If a lottery offers n distinct prizes and the

probabilities of winning the prizes are pi (i=1,…,n) then

1 21

... 1n

i ni

p p p p

An important concept: Expected Value

The expected value of a lottery is the average of the prizes obtained if we play the same lottery many times If we played 600 times the lottery in Example 2 We obtained a “1” 100 times, a “2” 100 times… We would win “£2” 300 times, win “£0” 200 times,

and lose “£6” 100 times Average prize=(300*2+200*0-100*6)/600 Average prize=(1/2)*2+(1/3)*0-(1/6)*6=0 Notice, we have the probabilities of the prizes

multiplied by the value of the prizes

Expected Value. Formal definition

For a lottery (X) with prizes x1,x2,…,xn and the probabilities of winning p1,p2,…pn, the expected value of the lottery is

1

( )n

i ii

E X p x

1 1 2 2( ) ... n nE X p x p x p x

The expected value is a weighted sum of the prizes the weights the respective probabilities

The symbol for the expected value of X is E(X)

Expected Value of monthly consumption (Example 3)

Example 3: John’s monthly consumption: X1=£4000 if he does not get ill

X2=£500 if he gets ill (so he cannot work) Probability of illness 0.25 Consequently, probability of no illness=1-0.25=0.75 The expected value is:

1 1 2 2( )E X p x p x ( ) 0.75*(£4000) 0.25(£500) 3125E X

Drawing the combinations of consumption with the same expected value

Only possible if we have at most 2 possible states (e.g. ill or not ill as in Example 3)

Given the probability p1 then p2=1-p1

How can we graph the combinations of (X1,X2) with a expected value of, say, “E”?


The combinations of (X1,X2) with an expected value of, say, “E”? 1 1 2 2

1 1 1 2

12 1

1 1

1 21

2 11

(1 )

1 1

0,1

0,

E p x p x

E p x p x

pEx x

p p

Eif x then x

p

Eif x then x

p


X1

X2

E/(1-p1)

E/p1

Decreasing line with slope:

-p1/(1-p1)

Introducing another lottery in John’s example

Lottery A: Get £3125 for sure independently of illness state (i.e. expected value= £3125). This is a lottery without risk

Lottery B: win £4000 with probability 0.75, and lose £500 with probability 0.25 (i.e. expected value also £3125)

Drawing the combinations of consumption with the same expected value. Example 3

X1

X2

3125/0.25

3125/0.75

Decreasing line with slope:

-p1/(1-p1) =-0.75/0.25=-3

Line of lotteries without risk

3125

3125

4000

500

Lot. A

Lot. B

A small game….

Lottery A: Get £3125 for sure (i.e. expected value= £3125)

Lottery B: Get £4000 with probability 0.75, and Get £500 with probability 0.25 (i.e. expected value also £3125)

Which one would you choose?

Is the expected value a good criterion to decide between lotteries?

One criterion to choose between two lotteries is to choose the one with a higher expected value

Does this criterion provide reasonable predictions? Let’s examine a case… Lottery A: Get £3125 for sure (i.e. expected value= £3125) Lottery B: get £4000 with probability 0.75, and get £500 with probability 0.25 (i.e. expected value also £3125)

Probably most people will choose Lottery A because they dislike risk (risk averse)

However, according to the expected value criterion, both lotteries are equivalent. The expected value does not seem a good criterion for people that dislike risk

If someone is indifferent between A and B it is because risk is not important for him (risk neutral)

Expected utility: The standard criterion to choose among lotteries

Individuals do not care directly about the monetary values of the prizes they care about the utility that the money provides

U(x) denotes the utility function for money We will always assume that individuals prefer more

money than less money, so:

'( ) 0iU x

Expected utility: The standard criterion to choose among lotteries

The expected utility is computed in a similar way to the expected value

However, one does not average prizes (money) but the utility derived from the prizes

The formula of expected utility is:

1 1 2 21

( ) ( ) ( ) ... ( )n

i i n ni

EU pU x pU x p U x p U x

The individual will choose the lottery with the

highest expected utility

Indifference curve

The indifference curve is the curve that gives us the combinations of consumption (i.e. x1 and x2) that provide the same level of Expected Utility

Drawing an indifference curve

Are indifference curves decreasing or increasing?

1 1 2 2

1 1 1 2

1 1 1 1 2 2

2 1 1

1 1 2

2

1

( ) ( )

( ) (1 ) ( )

0 '( ) (1 ) '( )

'( )*

(1 ) '( )

'( ) 0 0 sin

EU pU x p U x

EU pU x p U x

dEU pU x dx p U x dx

dx p U x

dx p U x

dxAsU x decrea g

dx

Drawing an indifference curve

Ok, we know that the indifference curve will be decreasing

We still do not know if they are convex or concave For the time being, let’s assume that they are convex If we draw two indifferent curves, which one represents

a higher level of utility? The one that is more to the right…

Drawing an indifference curveX2

X1

EU1

EU2

EU3

Convex Indifference curvesImportant to understand that:EU1 < EU2 < EU3


Indifference curve and risk aversion

X1

X2

3125/0.25

3125/0.75


3125

3125

4000

500

Lot. A

Lot. B

We had said that if the individual was risk averse, he will prefer Lottery A to Lottery B.

These indifference curves belong to a risk averse individual as the Lottery A is on an indifference curve that is to the right of the indifference curve on which Lottery B lies.

Lot A and Lot B have the same expected value but the individual prefers A because he is risk averse and A does not involve risk

Indifference curves and risk aversion

We have just seen that if the indifference curves are convex then the individual is risk averse

Could a risk averse individual have concave indifference curves? No….

Does risk aversion imply anything about the sign of U’’(x)

2 1 1

1 1 2

22 1 1

21 1 2

'( )*

(1 ) '( )

''( )*

(1 ) '( )

dx p U x

dx p U x

d x p U x

dx p U x

Convexity means that the second derivative is positive

In order for this second derivative to be positive, we need that U’’(x)<0

A risk averse individual has utility function with U’’(x)<0

What shape is the utility function of a risk averse individual?

X=money

U(x)

U’(x)>0, increasing U’’(x)<0, concave

Examples of commonly used Utility functions for risk averse individuals

( ) ln( )

( )

( ) 0 1

( ) exp( * ) 0

a

U x x

U x x

U x x where a

U x a x where a

Measuring Risk Aversion

The most commonly used risk aversion measure was developed by Pratt

"( )( )

'( )

U Xr X

U X

For risk averse individuals, U”(X) < 0 r(X) will be positive for risk averse

individuals

Risk Aversion

If utility is logarithmic in consumptionU(X) = ln (X )

where X> 0 Pratt’s risk aversion measure is

"( ) 1( )

( )

U Xr X

U X X

Risk aversion decreases as wealth increases

Risk Aversion

If utility is exponentialU(X) = -e-aX = -exp (-aX)

where a is a positive constant Pratt’s risk aversion measure is

2"( )( )

( )

aX

aX

U X a er X a

U X ae

Risk aversion is constant as wealth increases

Definition of certainty equivalent

The certainty equivalent of a lottery m, ce(m), leaves the individual indifferent between playing the lottery m or receiving ce(m) for certain.

In maths:

U(ce(m))=E[U(m)]

Definition of risk premium

Risk premium = E[m]-ce(m)

The risk premium is the amount of money that a risk-averse person would sacrifice in order to eliminate the risk associated with a particular lottery.

Intuition of risk premium

Assume that an individual can play the same lottery m many times

However, instead of each time playing the lottery, the individual plays safe and gets the ce(m)

The risk premium is the money that eventually the individual is losing by playing safe instead of playing the lottery

Risk premium in finance

(just in case you feel curious) In finance, the risk premium is the

expected rate of return above the risk-free interest rate.

Money

U(.)

Lottery m. Prizes m1 and m2

m1 m2

U(m1)

U(m2)

E[m]

E[U(m)]

ce(m)

Risk premium

Willingness to Pay for Insurance

Consider a person with a current wealth of £100,000 who faces a 25% chance of losing his automobile worth £20,000

Suppose also that the utility function is

U(X) = ln (x)


The person’s expected utility will be

E(U) = 0.75U(100,000) + 0.25U(80,000)

E(U) = 0.75 ln(100,000) + 0.25 ln(80,000)

E(U) = 11.45714


The individual will likely be willing to pay more than £5,000 to avoid the gamble. How much will he pay?

E(U) = U(100,000 - y) = ln(100,000 - y) = 11.45714

100,000 - y = e11.45714

y= 5,426

The maximum premium he is willing to pay

is £5,426

Actuarially fair premium If an agent buys an insurance policy at an actuarially fair

premium then the insurance company will have zero expected profits (note: marketing and administration expenses are not included in the computation of the actuarially fair premium)

Previous example: computing the expected profit of the insurance company:

EP=0.75paf + 0.25(paf-20,000) Compute paf such that EP=0. This is paf=£5000 Notice, the actuarially fair premium is smaller than the

maximum premium that the individual is willing to pay (£5426). This is a general results, risk averse individuals will be better off by if they are insured at the actuarially fair premium

Summary The expected value is an adequate criterion to choose

among lotteries if the individual is risk neutral However, it is not adequate if the individual dislikes risk

(risk averse) If someone prefers to receive £B rather than playing a

lottery in which expected value is £B then we say that the individual is risk averse

If U(x) is the utility function then we always assume that U’(x)>0

If an individual is risk averse then U’’(x)<0, that is, the marginal utility is decreasing with money (U’(x) is decreasing).

If an individual is risk averse then his utility function, U(x), is concave

A risk averse individual has convex indifference curves We have studied a standard measure of risk aversion The individual will insure if he is charged a fair premium

decision theory under uncertainty. uncertainty consumer and firms are usually uncertain about the...

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